Talk:Algebraic number theory/Archive 1

Archive 1

User:86.128.141.126

This needs to be corrected: "studying them modulo p for all primes p (see finite fields). This is called localization " 86.128.141.126 12:36, 4 March 2007 (UTC)

"Algebraic number theory" can also be interpreted as the arithmetic theory of algebraic numbers. Lhf 11:23, 11 October 2007 (UTC)

Organization of the article

Since it seems this may be the current COTM, I thought it would be good to suggest first a general structure to the article that could help frame future discussions. Here are some sections I can think of that would help organize the development of the article (I make no claim that this is complete or correct, please suggest additions/modifications):

• History
• Basic notions
• Subfields (aka Branches)
• Major results
• Open questions
• Relation to other fields

Thoughts? RobHar (talk) 23:46, 21 July 2008 (UTC)

really?

The article currently asserts, more or less, that algebraic number theory is the study of algebraic numbers. I thought it was more number theory (that is, the study of the naturals) using algebraic methods, as distinguished from analytic number theory. Of course the two things might be close to coextensive in practice, but we ought to get the concept right, as currently understood by practitioners in the field. (Clarification: I am not one of those practitioners.) --Trovatore (talk) 04:25, 24 July 2008 (UTC)

It can be taken in both senses, but I believe it's much more common to mean the theory of algebraic numbers (there is, for example, "Transcendental number theory"). Certain older texts on algebraic number theory use "theory of algebraic numbers" in their title (such as Hecke's classic "Lectures on the Theory of Algebraic Numbers", and Ribenboim's "Classical Theory of Algebraic Numbers", both of whose content is pretty standard for an "algebraic number theory" book). Another example is the preface of Neukirch's "algebraic number theory" which consistently refers to the content of the book as being about the study of algebraic numbers. This meaning also appears to be the meaning that springer's encyclopedia of mathematics and planetmath both take. A lot of analytic number theory consists of analytic results about algebraic numbers. And a lot of algebraic number theory uses analytic methods such as automorphic forms, p-adic analysis, p-adic functional analysis to name a few. I think algebraic number theory is defined by the problems it seeks to answer rather than by the methods it uses to answer them, is perhaps a good way to put it. RobHar (talk) 06:22, 24 July 2008 (UTC)

I'm really not sure, and I might myself be an algebraic number theorist (I'm so unsure that I'm unsure about that too!). To my mind, "the theory of algebraic numbers" is a big part of algebraic number theory but not necessarily synonymous with it. Were I to have to give a one sentence definition, I might try: "Algebraic number theory is the study of local and global fields, their associated rings of integers, and other associated structures and invariants, especially the class groups, unit groups and Galois groups." Of course that's much harder for the reader who does not already know some algebraic number theory to understand.

It may not be necessary or profitable to give absolutely the right definition of algebraic number theory before working on the article. I think that Rob (Hi, Rob!) Harron's outline is a good step 0; step 1 (the hard part!) will be to fill in some of these things. I'm not sure yet whether I will participate in this or not -- it might be too ambitious for me. Plclark (talk) 19:53, 25 July 2008 (UTC)Plclark

That is you isn't it? Anyway, I was actually in the process of altering the intro while you were leaving this message, and I think I may have been able to provide a vague enough description for it to be pretty much true, but not too vague as to be completely useless. I said it's a "branch of number theory which studies the algebraic structures related to algebraic integers". I definitely agree that this should only be a first step, and that the description should develop as more work is put in and more people contribute. And if you are who I think you are, plclark, then ambition in writing is right up your alley and you should definitely participate (at least a bit...) RobHar (talk) 20:21, 25 July 2008 (UTC)

References

I find the statement about the "only classic text still in print [...]" to be somewhat strange. Books like Neukirch's Algebraic Number Theory (ISBN 3540653996) come to mind, and is still available. Particular topics missing? Moreover, the reference sections should not be limited to books currently in print. Stca74 (talk) 15:20, 5 August 2008 (UTC)

Indeed, I believe that section was just pretty much copied over from the analytic number theory article by someone, then I made some changes to make it more relevant, but I do think the reference section as it is should go. But I figured it should be left there until I was (or somebody) going to replace it with another section of references. Perhaps for now I'll quickly adjust it to make it better. But I would hope that would only be a temporary measure as this is currently the (much underworked) collaboration of the month. Though I do think the "only classic" comment is indeed strange for a reference section, it is pretty much accurate (though one could cite Weil's "Basic number theory"). Lang's book was originally published in 1970 as an expansion of a 1964 book, whereas Neukirch's original german edition was published in 1992.

If you have any suggestions of good books that could be included in a nice references (or further reading) section, that would be nice. I'm mainly aware of graduate level texts. Thanks for the prod. RobHar (talk) 17:43, 5 August 2008 (UTC)

The books I thought about and have used most (Fröhlich-Taylor, Neukirch) are now already listed. Perhaps could add Serre's Corps locaux, even though its focus is obviously on the local fields side of things. And I suppose Weil's Basic Number Theory could deserve a mention, although the harmonic analysis approach takes one in slightly different direction. Stca74 (talk) 18:57, 9 August 2008 (UTC)